Means and Extremes

Explaining how to solve a proportion by cross-multiplying is pretty straightforward. It's just a procedure. Explaining why the product of the means and the product of the extremes are equal is a bit trickier, but actually really simple. Warning, though: It's not a sexy explanation by any stretch.

To review a bit, we'll talk about a proportion as an equation showing that two ratios are equivalent. For example, 60 miles per hour is the exact same speed as 120 miles every 2 hours. The ratio (which is a rate in this case) 60 mi : 1 h is equivalent to the rate 120 mi : 2 h, and we can write this as a proportion:

60 mi : 1 h = 120 mi : 2 h

Writing the proportion this way helps you understand why we use the terms means and extremes. The two middle numbers are the means, and the two outer numbers are the extremes.

To make sense of the term cross-multiply, though, you should write this proportion using fraction notation:

So, we can cross-multiply the means (1 × 120) and the extremes (60 × 2) and see that the respective products are equal (1 × 120 = 120, and 60 × 2 = 120). But why are they equal?

The trick is pretty simple. Use inverse operations to rewrite the proportion so that the "fractions" disappear. We can start anyway we like, but here let's first multiply both sides of the equation by 2 to get rid of that "fraction" on the right side of the equation:

And now we just multiply both sides by 1 to get rid of the "fraction" on the left side:

And there you have it! The best way to see that this works for all proportions is to replace the numbers with variables and do the same thing we did above: